Honors Geometry Sections 10.1 & 10.2
Trigonometric ratios
The word “trigonometry” comes from two
Greek words which mean
___________________
triangle measurement
And that is exactly what we will do with
trigonometry - we will find the measures of
angles and the length of sides in triangles.
Initially, we will consider only right
triangles, but later in this section, we will
consider trigonometry involving acute and
obtuse triangles.
A trigonometric ratio (i.e. fraction) is a
ratio of the lengths of two sides of a
right triangle. The three basic
trigonometric ratios are sine (___),
sin
cosine (___)
cos and tangent (___).
tan These
ratios are defined for the acute angles
of a right triangle as follows.
opposite leg
sin A = -------------hypotenuse
adj.
hyp.
opp.
adjacent leg
cos A = -------------hypotenuse
opposite leg
tan A = adjacent
------------leg
If we write the ratios for the acute
angle B instead, then the opposite
leg and adjacent leg would be
switched!!!!
An easy way to remember these
three trig ratios is with the
mnemonic
SOH-CAH-TOA
A P D
I P Y
O D Y
N P
E
P
S J
P
N P J
Example: Give the three trig ratios for the
following triangles.
hyp
opp
adj
3
5
4
5
3
4
Example: Give the three trig ratios for the
following triangles.
5 5
10 10 5 2 5


Sin O =
25
5
5 5
10 10 5 2 5


Cos G =
25
5
5 5
5 1
Tan G = 
10 2
5  10  c
2
2
2
c  125
2
c  125  5 5
The value of the sine, cosine and tangent of an
angle depend only upon the measure of the
angle and not the size of the triangle that the
angle is found in. Here’s why!!!
ACE ~ BCD by _____
AA
Therefore, AC  AE  CE
BC
BD
CD
ACE ~ BCD by _____
AA
Therefore, AC 
BC
AE CE

BD CD
Now, we can take the first two ratios separately and
AE

rearrange the terms to get BD
. Note: These are the
BC AC
ratios for sin C in both ACE and BCD .
Similarly, we can take the first and third ratios
CD CE
separately and rearrange the terms to get BC  AC . Note:
These are the ratios for cos C in both ACE and BCD .
A scientific calculator can be used to find
the value of these three trig ratios. Make
sure your calculator is in degree mode.
.225
sin 130 = ___________
.225
tan 400 = ____________
.839
cos 770 = ____________
We can use the trig ratios to find
the lengths of unknown sides in
right triangles.
Example: Solve for x and y. Round your answers
to the nearest 1000th.
12
sin 28 
x
x sin 28  12
12
x
sin 28
x  25.561
tan 28 
12
y
y tan 28  12
12
y
tan 28
y  22.569
Example: Solve for x and y. Round your answers
to the nearest 1000th.
x
cos 40 
20
20 cos 40  x
x  15.321
y
sin 40 
20
20 sin 40  y
y  12.856
Example: The angle of elevation to the top
of a tree from a point 100 feet from the
base of the tree is 510. Estimate the height
of the tree to the nearest 1000th.
NOTE: the angle of elevation is the angle formed
by a horizontal line (usually the ground) and the
line of sight up to some object.
Example: The angle of elevation to the top
of a tree from a point 100 feet from the
base of the tree is 510. Estimate the height
of the tree to the nearest 1000th.
x
tan 51 
100
100 tan 51  x
x  123.490 ft
x
51
100